Next step is to create new user whose role is Ranking Editor.
This user can login well, and he will see dashboard, profile, and newly created Ranking menu position. User can enter "Ranking settings", but trying to save anything inside gets:
"Cheatin' eh?"

The only solution for this problem is to add:

$role->add_cap('manage_options');

in first step, but this allows user to edit options, and that is not how problem should be fixed.
This is the only option which works, that custom role must have, so it can edit custom admin menu page that have this role attached to.